高频散射问题和传输特征值的数值求解

The numerical calculation of high frequency scattering problems and transmission eigenvalues is of important in the fields of scientific computation.It is also an international recognized difficulties and hot research topics.These problems arise in the study of the direct and inverse scattering problems.These problems have great theoretical significance and wide application values.We mainly study the theory and numerical approximation for the high frequency scattering problems and transmission eigenvalues in acoustic and electromagnetic scattering problems.We shall consider in this project the following issues.1.We study the oscillatory behaviors and fast solver technologies to the solution of the differential equations and integral equations in the high frequency scattering problems.2.We study the numerical methods for solving the transmission eigenvalues.The expected results of this project would not only enrich the mathematical theory and numerical computation of the direct and inverse scattering problems, but also provide some valuable assistance and reference for the numerical computation to other high oscillation problems and the design of numerical methods to the inverse scattering problems.These results are important to scientific research and practical applications.

高频散射问题和传输特征值的数值计算是科学计算领域非常重要的,国际公认的难点与热点研究课题,广泛的应用在正、反散射问题的研究中,具有深远的理论意义和重要的应用价值.本项目主要研究声学与电磁学中,高频散射问题和传输特征值问题的理论和高效数值计算方法.包括以下方面的内容:1.高频散射问题中微分方程和积分方程解的振荡性态,和快速求解技术.2.传输特征值的高效数值求解方法.本项目的预期研究成果,不仅能丰富正、反散射问题的数学理论和高效数值算法,也将为研究其它振荡问题的数值计算和反散射问题数值算法的设计提供有益的参考与帮助,在科学研究和实际应用中有着重要的现实意义.

微分方程特征值问题和高频散射问题的研究是由实际应用驱动，相关问题的理论分析和数值计算具有理论意义和潜在应用价值，本项目围绕声学与电磁学中的特征值问题和高频散射问题，展开了以下研究：（1）传输特征值和Steklov特征值的数值求解；（2）具有奇性和振荡积分核的积分方程的理论分析和数值计算；（3）求解Helmholtz方程边值问题的深度学习方法。上述研究内容不仅可以丰富散射问题的数学理论，同时为发展高效的新型数值方法提供有益参考。